1. Field of the Invention
The invention relates to methods and apparatus for assessing the reflective properties of mirrors at different angles of incidence without precise knowledge of the mirror's basic optical constants and/or without precise knowledge of the mirror's over-coating prescription. More particularly, the invention relates to accurately estimating reflectance values for multiple angles of incidence based upon measured reflectance values collected for a single or a few angles of incidence.
2. Description of the Related Art
Many remote sensing instruments make use of mirrors that scan over a wide range of angles to produce a large field-of-view image, such as those images typically taken by satellites of the earth, or images taken by optical telescopes of distant planets and galaxies. While this approach is simple in concept, a significant complication is that the reflectance of a scanning mirror varies, depending upon the angle at which light strikes the mirror (i.e., the “angle of incidence” of the incoming light). This angle of incidence of incoming light is typically measured as an angle relative to a perpendicular to the surface of the mirror at the point of incidence.
A further complication is that the reflectance of the S-polarization and P-polarization components of light at a particular wavelength vary differently as the angle of incidence changes. Unless corrections are made to account for the changes in a scanning mirror's S-polarization reflectance and P-polarization reflectance as a function of the angle of incidence, the uncorrected changes will decrease the calibration accuracy of the remote sensing instrument's output measurements. As a consequence, it is necessary to produce a calibration equation that accurately includes a model of the instrument's response as a function of angle of incidence. Such a model is preferably based on a theoretical mathematical model of the underlying physical reflectance phenomenon, or, if that is not possible, a combination of a physical model and measured reflectance values.
For most remote sensing mirrors, calibration uncertainties associated with scan mirror reflectance are made more serious by the use of absorbing coatings on the mirror surfaces to protect them from damage. While these coatings provide protection from damage, they also change the angular dependence of the reflectance of the mirror, particularly in some key wavelength ranges of the IR spectrum. If the optical constants of the substrate and the over-coating are accurately known, reflectance of the mirror at different angles of incidence is easily calculated using the basic Fresnel equations. The difficulty for instrument manufacturers is that the over-coating prescription is not necessarily known (e.g., proprietary coatings are used) or the manufacturing and storage process may not be controlled well enough to allow accurate calculation of the reflectance from basic optical constants of the deposited protective material. Furthermore, the optical qualities of over-coatings can change over time, due to prolonged exposure to extreme conditions, requiring that the reflectivity of a mirror be periodically reassessed for recalibration purposes.
Devices that use a scanning mirror to take measurements typically adjust individual measurements made by the device to accommodate variances in the reflectance of the scanning mirror for the wavelength and specific angle of incidence at which the measurement was made. To assure maximum accuracy of measurements the reflective properties of the mirror at different angles of incidence are, preferably, periodically recalculated. There are several ways in which such calibration/recalibration can be achieved.
First, with knowledge of the indices of refraction (n) and the absorption coefficients (k) of the coated mirror layers, the Fresnel reflectance equations can be used as a physical model to calculate expected reflectance values. However, values of n and k vary widely as a function of the wavelength of the light (λ) and the details of the mirror coating process. Therefore, calculated reflectance values are preferably compared against measured reflectance values to enhance confidence and/or to develop linear scaling factors to accommodate external or unknown physical effects not represented in the Fresnel equations. Once validated, the corrected Fresnel equations can be used to extrapolate highly accurate reflectance values. However, as previously discussed, this approach is typically hindered by a lack of information with respect to proprietary mirror coatings, and/or changes in optical characteristics due to environmental effects.
As a second approach, a set of reflectance values are physically measured at a specific wavelength and used to develop a polynomial equation that “fits” the measured mirror reflectance values as a function of angle of incidence for each wavelength. The polynomial equation developed for each wavelength can then be used as the model/algorithm for calculating the angle of incidence effect on instrument throughput for the wavelength so modeled. Although such polynomial equations are able to fit measured reflectance values to a rather high degree of precision, a polynomial equation is not based upon a mathematical model of the physical phenomenon of mirror reflectance. Therefore, a polynomial equation does not fundamentally provide a true representation of the expected behavior of the mirror as a function of angle of incidence. The polynomial equation merely connects the measured reflectance values irrespective of whether the measured reflectance values contain errors. Values of reflectance calculated using such a polynomial equation will, therefore, inherently integrate previous measurement errors. Errors in the measured reflectance values to which the polynomial equation is fitted will not be detected, and reflectance values calculated using the resulting polynomial equation will be consistent with the original faulty reflectance value measurements.
Another deficiency associated with the use of polynomial equations to model mirror reflectance is that the approach requires a significant number of manual measurements of mirror reflectance versus angle of incidence so that an accurate polynomial equation can be developed that matches the resulting measured reflectance values. Such measurements must be taken for each wavelength with which the mirror will be used, because previously developed polynomial equations are not based upon a physical model and, therefore, cannot be accurately relied upon to predict reflectivity of the mirror at other wavelengths.
In a third approach, a large number of physical measurements are made and stored in one or more lookup tables. Reflectance measurements must be made for each angle of incidence/wavelength combination at which the scan mirror is used. Interpolation is then used to extrapolate values between measured points. Although simple in concept, such an approach requires a large number of measured reflectance value data points and sufficient storage capacity to support use of the technique. Recalibration of devices that use such a technique requires taking an entirely new set of reflectance value measurements, specific to each individual device.
While all three approaches can be used to extrapolate the reflectance values of a coated mirror in a laboratory (i.e., with the assistance of special test equipment and special test configurations), all three approaches suffer from serious deficiencies with respect to their ability to be adapted for use in deployed devices (i.e., outside of a controlled laboratory environment in which specialized support equipment and personnel are not readily available). Furthermore, these approaches cannot be used to develop high confidence reflectance values at multiple angles of incidence based upon reflectance values measured at one or a few angles of incidence, thereby significantly increasing the number of measured reflectance values required to facilitate generation of accurate reflectance values across multiple angles of incidence. Such deficiencies are further exacerbated in remote sensing devices deployed into orbit around the earth or deployed to the far reaches of outer-space. Such devices are, typically, constrained in size, weight and design and are inaccessible to technicians with the required calibration equipment.
Hence, there is a need for methods and apparatus capable of accurately estimating the angular dependence of reflectivity of a coated mirror across a broad spectrum of wavelengths while based upon a reduced number of manual measurements and without precise knowledge of the mirror's basic optical constants and/or without precise knowledge of the mirror's over-coating prescription. The new methods and apparatus should integrate physical models of the angular dependence of mirror reflectance that can be used to validate reflectivity measurements of a mirror at different angles of incidence and wavelengths despite variances in the refraction and absorption qualities of the mirror's protective coating. The novel methods and apparatus should simplify present physical reflectance models so that accurate reflectance values can be quickly and efficiently extrapolated, so that errors associated with measured mirror reflectance values at different angles of incidence can be quickly identified, and measurements made from light reflected by a scanning mirror can be readily corrected.